DLMF: Untitled Document

… ►Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients

\genfrac{(}{)}{0.0pt}{}{m}{n}

for

m

up to 50 and

n

up to 25; extends Table 26.4.1 to

n=10

; tabulates Stirling numbers of the first and second kinds,

s\left(n,k\right)

and

S\left(n,k\right)

, for

n

up to 25 and

k

up to

n

; tabulates partitions

p\left(n\right)

and partitions into distinct parts

p\left(\mathcal{D},n\right)

for

n

up to 500. ►Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts

\not\equiv\pm 2\pmod{5}

, partitions into parts

\not\equiv\pm 1\pmod{5}

, and unrestricted plane partitions up to 100. … ►Goldberg et al. (1976) contains tables of binomial coefficients to

n=100

and Stirling numbers to

n=40

.

… ►

25.10.1

Z(t)\equiv\exp\left(i\vartheta(t)\right)\zeta\left(\tfrac{1}{2}+it\right),

ⓘ

Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\equiv$ : equals by definition, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $Z(t)$ : zeros function and $\vartheta(t)$ : function
Keywords:: definition
Source:: Titchmarsh (1986b, (4.17.2), p. 89)
Permalink:: http://dlmf.nist.gov/25.10.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §25.10(i), §25.10 and Ch.25

… ►

25.10.2

\vartheta(t)\equiv\operatorname{ph}\Gamma\left(\tfrac{1}{4}+\tfrac{1}{2}it% \right)-\tfrac{1}{2}t\ln\pi

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$ : complex conjugate, $\equiv$ : equals by definition, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase, $z$ : complex variable, $\vartheta(t)$ : function and $\chi(s)$ : function
Keywords:: definition
Proof sketch:: Derivable from Titchmarsh (1986b, third equation on p. 89), namely $\vartheta(t)\equiv-\frac{1}{2}\operatorname{ph}\chi(\frac{1}{2}+\mathrm{i}t)$ , by using (25.9.2), (1.9.18), (1.9.20), $\Gamma\left(\overline{z}\right)=\overline{\Gamma\left(z\right)}$ , and (4.8.10).
Permalink:: http://dlmf.nist.gov/25.10.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §25.10(i), §25.10 and Ch.25

… ►More than 41% of all the zeros in the critical strip lie on the critical line (Bui et al. (2011)). …

… ►where

m\equiv n\not\equiv 0\pmod{p-1}

. …valid when

m\equiv n\pmod{(p-1)p^{\ell}}

and

n\not\equiv 0\pmod{p-1}

, where

\ell(\geq 0)

is a fixed integer. … ►

24.10.8

N_{2n}\equiv 0\pmod{p^{\ell}},

ⓘ

Symbols:: $\equiv$ : modular equivalence, $\ell$ : integer, $n$ : integer and $p$ : prime
Referenced by:: §24.10(iv)
Permalink:: http://dlmf.nist.gov/24.10.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §24.10(iv), §24.10 and Ch.24

►valid for fixed integers

\ell(\geq 1)

, and for all

n(\geq 1)

such that

2n\not\equiv 0

\pmod{p-1}

and

p^{\ell}\mathbin{|}2n

. ►

24.10.9

E_{2n}\equiv\begin{cases}0\pmod{p^{\ell}}&\text{if }p\equiv 1\pmod{4},\\ 2\pmod{p^{\ell}}&\text{if }p\equiv 3\pmod{4},\end{cases}

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\equiv$ : modular equivalence, $\ell$ : integer, $n$ : integer and $p$ : prime
Referenced by:: §24.10(iv)
Permalink:: http://dlmf.nist.gov/24.10.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §24.10(iv), §24.10 and Ch.24

…

… ►

§27.14(v) Divisibility Properties

►Ramanujan (1921) gives identities that imply divisibility properties of the partition function. For example, the Ramanujan identity …implies

p\left(5n+4\right)\equiv 0\pmod{5}

. …For example,

p\left(1575\;25693n+1\;11247\right)\equiv 0\pmod{13}

. …

… ►

J. V. Wehausen and E. V. Laitone (1960) Surface Waves. In Handbuch der Physik, Vol. 9, Part 3, pp. 446–778.

ⓘ

External Links:: Online edition, MathReview (F. Ursell), ZentralBlatt
Referenced by:: §11.12.
Encodings:: BibTeX

… ►

S. W. Weinberg (2013) Lectures on Quantum Mechanics. Cambridge University Press, Cambridge, UK.

ⓘ

External Links:: ISBN 978-1-107-02872-2
Referenced by:: §18.39(ii).
Encodings:: BibTeX

… ►

H. S. Wilf and D. Zeilberger (1992a) An algorithmic proof theory for hypergeometric (ordinary and “

q

”) multisum/integral identities. Invent. Math. 108, pp. 575–633.

ⓘ

External Links:: ISSN 0020-9910, Document, MathReview (David R. Masson), ZentralBlatt
Referenced by:: §16.4(iii).
Encodings:: BibTeX

►

H. S. Wilf and D. Zeilberger (1992b) Rational function certification of multisum/integral/“

q

” identities. Bull. Amer. Math. Soc. (N.S.) 27 (1), pp. 148–153.

ⓘ

External Links:: ISSN 0273-0979, Pdf, Document, MathReview (Robert A. Gustafson), ZentralBlatt
Referenced by:: §16.4(iii).
Encodings:: BibTeX

…

… ►Thus,

y\equiv x^{r}\pmod{n}

and

1\leq y<n

. … ►By the Euler–Fermat theorem (27.2.8),

x^{\phi\left(n\right)}\equiv 1\pmod{n}

; hence

x^{t\phi\left(n\right)}\equiv 1\pmod{n}

. But

y^{s}\equiv x^{rs}\equiv x^{1+t\phi\left(n\right)}\equiv x\pmod{n}

, so

y^{s}

is the same as

x

modulo

n

. …

… ►That 1046-page tome proved to be an invaluable reference for the many scientists and engineers who use the special functions of applied mathematics in their day-to-day work, so much so that it became the most widely distributed and most highly cited NIST publication in the first 100 years of the institution’s existence. … ►The online version, the NIST Digital Library of Mathematical Functions (DLMF), presents the same technical information along with extensions and innovative interactive features consistent with the new medium. …

… ►

F. Oberhettinger and L. Badii (1973) Tables of Laplace Transforms. Springer-Verlag, Berlin-New York.

ⓘ

Notes:: Table errata: Math. Comp. v. 50 (1988), no. 182, pp. 653–654.
External Links:: ISBN 0-387-06350-1, MathReview, ZentralBlatt
Referenced by:: §1.14(viii), §10.22(vi), §10.43(vi), §11.5(iii), §11.7(v), §11.9(iv), §12.12, §12.5(iv), §13.10(ii), §13.23(i), §14.17(v), §15.14, §18.17(ix), §20.10(ii), §5.13, §6.14(iii), §6.7(iv), §7.14(iii), §7.7(iii), §8.6(iii).
Encodings:: BibTeX

… ►

T. Oliveira e Silva (2006) Computing

\pi(x)

: The combinatorial method. Revista do DETUA 4 (6), pp. 759–768.

ⓘ

Notes:: Online fulltext contains postpublication annotations.
External Links:: FullText
Referenced by:: §27.18.
Encodings:: BibTeX

… ►

I. Olkin (1959) A class of integral identities with matrix argument. Duke Math. J. 26 (2), pp. 207–213.

ⓘ

External Links:: Document, MathReview (R. Bellman), ZentralBlatt
Referenced by:: §35.3(ii).
Encodings:: BibTeX

…

… ►

Table 26.6.3: Narayana numbers

N(n,k)

.

► ►►►

$n$	$k$
$n$	…
6	0	1	15	50	50	15	1
…

► … ►

§26.6(iv) Identities

►

26.6.12

C\left(n\right)=\sum_{k=1}^{n}N(n,k),

ⓘ

Symbols:: $C\left(\NVar{n}\right)$ : Catalan number, $k$ : nonnegative integer, $n$ : nonnegative integer and $N(n,k)$ : Narayana number
Permalink:: http://dlmf.nist.gov/26.6.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §26.6(iv), §26.6 and Ch.26

…

… ►

G. M. D’Ariano, C. Macchiavello, and M. G. A. Paris (1994) Detection of the density matrix through optical homodyne tomography without filtered back projection. Phys. Rev. A 50 (5), pp. 4298–4302.

ⓘ

External Links:: Document
Referenced by:: §13.28(iii).
Encodings:: BibTeX

… ►

K. Dilcher, L. Skula, and I. Sh. Slavutskiǐ (1991) Bernoulli Numbers. Bibliography (1713–1990). Queen’s Papers in Pure and Applied Mathematics, Vol. 87, Queen’s University, Kingston, ON.

ⓘ

Notes:: A frequently updated version, with links to reviews, exists online
External Links:: Online version, MathReview, ZentralBlatt
Referenced by:: Ch.24.
Encodings:: BibTeX

… ►

B. I. Dunlap and B. R. Judd (1975) Novel identities for simple

n

-

j

symbols. J. Mathematical Phys. 16, pp. 318–319.

ⓘ

External Links:: Document, MathReview (M. J. Westwater)
Referenced by:: §34.5(vi).
Encodings:: BibTeX

… ►

P. L. Duren (1991) The Legendre Relation for Elliptic Integrals. In Paul Halmos: Celebrating 50 Years of Mathematics, J. H. Ewing and F. W. Gehring (Eds.), pp. 305–315.

ⓘ

External Links:: ISBN 0-387-97509-8, MathReview (J. M. H. Peters), ZentralBlatt
Referenced by:: §19.7(i).
Encodings:: BibTeX

…

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1—10 of 204 matching pages

1: 26.21 Tables

2: 25.10 Zeros

3: 24.10 Arithmetic Properties

4: 27.14 Unrestricted Partitions

§27.14(v) Divisibility Properties

5: Bibliography W

6: 27.16 Cryptography

7: Foreword

8: Bibliography O

9: 26.6 Other Lattice Path Numbers

§26.6(iv) Identities

10: Bibliography D